Intereting Posts

What's so special about the 4 fundamental subspaces?
What does an outer automorphism look like?
Is there a reason to prefer the improper integral for some cases over the Cauchy principal value?
Conjectured primality test
Fermats Little Theorem
Prove $\sum_{k=1}^{\infty} \frac{\sin(kx)}{k} $ converges
e as sum of an infinite series
Making a cube root function analytic on $\mathbb{C}\backslash $
What is the proof that the total number of subsets of a set is $2^n$?
Can we write every uncountable set $U$ as $V∪W$, where $V$ and $W$ are disjoint uncountable subsets of $U$?
Computing the Integral $\int r^2 \text{J}_0(\alpha r) \text{I}_1(\beta r)\text{d}r$
Prove that if |G|=132 then G cannot be simple
What's the difference between $\mathbb{Q}$ and $\mathbb{Q}(\sqrt{-d})$?
Rigorous nature of combinatorics
Proving that $\frac{e^x + e^{-x}}2 \le e^{x^2/2}$

**The question goes like this-**

How many strings can be generated by permuting the characters of “**abbbbcccdeff**” such that there are only 3 mismatchings and the rest 9 are same ?

**My attempt-**

- How many combinations can be made with these rules? (game of Dobble)
- Interpretation of a combinatorial identity
- In how many ways can we put $31$ people in $3$ rooms?
- How many different possible permutations are there with k digits that add up to n?
- how many unique patterns exist for a NxN grid
- Probability of men and women sitting at a table alternately

Obviously, If the string was having only distinct characters(like-“**abcdefghijkl**“) then the answer would have been- 2*(12C9)=440 strings[as there are 12 characters and 9 of them have to be same]. I can calculate this for strings having distinct characters only, but I am failing to generalise this for strings with repeating characters.

I can manually find all pairs,but this method is very time-consuming,like-for the case of- **“abbbcc”** there comes a total of 12 such strings . These are-

**bbbcac**,**bbbcca**,**bbcbac**,**bbcbca**,**bcbbac**,**bcbbca**,**cabbbc**,**cabbcb**,**cbabbc**,**cbabcb**,**cbbabc**,**cbbacb**

**Where I am failing?**

I need a quick solution(some kind of generalised formula) instead of counting it manually

- The $5n+1$ Problem
- Combinatorial interpretation of an alternating binomial sum
- Can an 8×8 square be tiled with these smaller squares?
- Number of permutations of $n$ where no number $i$ is in position $i$
- Combinatorial Argument for Recursive Formula
- Maximum number of edges that a bipartite graph with $n,m$ vertices can have when it doesn't contain $4$-cycle
- Generating a Eulerian circuit of a complete graph with constant memory
- Dimension of the space of algebraic Riemann curvature tensors
- Combinatorial proof that $\sum \limits_{k=0}^n \binom{2k}{k} \binom{2n-2k}{n-k} (-1)^k = 2^n \binom{n}{n/2}$ when $n$ is even
- Derangement of n elements

There is $1$ letter with $4$ occurrences, $1$ letter with $3$ occurrences, $1$ letter with $2$ occurrences and $3$ letters with $1$ occurrence. Thus the number of triples of distinct letters is

$$

\binom30(4\cdot3\cdot2)+\binom31(4\cdot3\cdot1)+\binom31(4\cdot2\cdot1)+\binom31(3\cdot2\cdot1)+\binom32(4+3+2)\cdot1+\binom33=24+36+24+18+27+1=130\;.

$$

Each of these can be permuted in $2$ ways, for a total of $2\cdot130=260$ permutations.

- Accumulation Points for $S = \{(-1)^n + \frac1n \mid n \in \mathbb{N}\}$
- What is this automorphism-related subgroup?
- Power summation of $n^3$ or higher
- Compute $\mathrm{Tor}_{n}^{\mathbb{Z}_{8}}(\mathbb{Z}_{4},\mathbb{Z}_{4})$
- Diagonal $\Delta = \{x \times x : x \in X \}$ closed in $X \times X$ implies that $X$ is Hausdorff
- Lunch Meeting Probability for two person to meet in given 1 hour slot and none would wait more then 15 minute.
- What does this quotient group $\frac{C^*}{R^+}$ represent?
- Divergent or not series?
- $\# \{\text{primes}\ 4n+3 \le x\}$ in terms of $\text{Li}(x)$ and roots of Dirichlet $L$-functions
- $A_4$ has no subgroup of order $6$?
- How to derive an identity between summations of totient and Möbius functions
- Historic proof of the area of a circle
- Fibonacci number identity.
- How to calculate the distance between this two houses?
- Finding the order of permutations in $S_8$